FAD1014 Tonight Full Drill — SOLUTIONS ONLY

Do NOT look at this until you have attempted all questions.

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SECTION A: INTEGRATION — SOLUTIONS

Solutions A1–A4

Q1

(a) $\boxed{x^3 - 2x^2 + 5x + C}$

(b) $\boxed{2\ln|x| + C}$

(c) $\frac{1}{4}e^{4x} + C$

(d) $-\frac{1}{3}\cos(3x) + C$

Q2

(a) $\frac{1}{3}(x^2+1)^{3/2} + C$

(b) $\frac{1}{2}\ln(x^2+4) + C$

(c) $\frac{1}{3}e^{x^3} + C$

Q3

(a) $1$

(b) $\frac{1}{8}$

(c) $\frac{1}{2}$

Q4

(a) $xe^x - e^x + C$

(b) $\frac{x^3}{3}\ln x - \frac{x^3}{9} + C$

(c) $-x\cos x + \sin x + C$

Q5

(a) $x\ln x - x + C$

(b) $\frac{e^x(\sin x - \cos x)}{2} + C$

(c) $e - 2$

Q6

(a) $\sin^{-1}\left(\frac{x}{2}\right) + C$

(b) $\frac{1}{3}\ln|3x + \sqrt{1+9x^2}| + C$

(c) $\ln|x + \sqrt{x^2-9}| + C$

Q7

(a) $\frac{\pi}{6}$

(b) $-\frac{\sqrt{x^2+4}}{4x} + C$

Q8

$$\frac{26}{3}$$

Q9

$$1$$

Q10

$$2$$

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SECTION B: DIFFERENTIAL EQUATIONS — SOLUTIONS

Solutions B1–B4

Q11–Q15 ID

Q	Type	Method
11	Separable	$y,dy = x^2,dx$
12	Homogeneous	$y = vx$, all terms degree 2
13	Separable	$\frac{dy}{1+y^2} = 2x,dx$
14	Non-Homogeneous (Dependent)	$\frac{1}{2} = \frac{2}{4}$, substitute $z = 2x + y$
15	Non-Homogeneous (Independent)	$\frac{2}{1} \neq \frac{1}{-1}$, solve for $h,k$

Q16

$$y = 2e^{x^2/2}$$

Q17

$$\frac{y^3}{3} + y = \frac{x^2}{2}$$

Q18

$$y = x$$

Q19

$$\ln|y| + \frac{y^2}{2} = \sin x + \frac{1}{2}$$

Q20

$$y = 2e^{(x^3-1)/3} - 1$$

Q21

$$y^2 = 2x^2(\ln|x| + C)$$

Q22

$$y^2 = x^2(2\ln|x| + 1)$$

Q23

$(x+y)^{1/2} = C|x-y|^{3/2}$ (implicit form)

Q24

(a) Dependent: $\frac{2}{4} = \frac{1}{2}$

(b) Substitute $z = 2x + y$ → separable: $\frac{2z}{3} + \frac{5}{9}\ln|3z-7| = x + C$

Q25

(a) Independent: $\frac{1}{1} \neq \frac{-1}{1}$

(b) Simultaneous: $x = 1$, $y = 3$. Substitute $X = x-1$, $Y = y-3$ → homogeneous.

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SECTION C: SERIES & SUMMATION — SOLUTIONS

Solutions C1–C4

Q26

(a) Converges to $\frac{3}{5}$

(b) Diverges ($\to \infty$)

(c) Diverges (oscillates)

Q27

(a) Converges ($p$-series, $p=2 > 1$)

(b) Diverges (nth term $\to 1 \neq 0$)

(c) Diverges ($r = 3/2 > 1$)

Q28

$$\frac{n}{n+1}$$

Q29

$$\frac{3}{2} - \frac{1}{n+1} - \frac{1}{n+2}$$

Q30

$$610$$

Q31

$$\frac{n^2 + 3n + 1}{4(n+1)(n+2)}$$

Q32

(a) $16x^4 + 32x^3 + 24x^2 + 8x + 1$

(b) $x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32$

(c) $16x^4 - 32x^2 + 24 - \frac{8}{x^2} + \frac{1}{x^4}$

Q33

(a) $160$

(b) $-270$

(c) $240$

Q34

(a) $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

(b) $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

(c) $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

(d) $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$

Q35

(a) $1 - 2x + 2x^2 - \frac{4x^3}{3} + \frac{2x^4}{3} + \cdots$

(b) $x^2 - \frac{x^6}{6} + \cdots$ (up to $x^4$: $x^2$)

(c) $1 - \frac{9x^2}{2} + \frac{27x^4}{8} + \cdots$

(d) $2x - 2x^2 + \frac{8x^3}{3} - 4x^4 + \cdots$

Q36

(a) $x^2 + x^3 + \frac{x^4}{2} + \cdots$

(b) $x - \frac{x^3}{2} + \frac{x^5}{24} - \cdots$

(c) $x + x^2 + \frac{x^3}{3} + \cdots$

Q37

$0.461$ (3 d.p.)

Q38

$0.0998$ (4 d.p.)

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SECTION D: GEOMETRY — SOLUTIONS

Solutions D1–D4

Q39

(a) Vertex $(2,-1)$, Focus $(2,1)$, Directrix $y = -3$, Axis $x = 2$

(b) Vertex $(3,0)$, Focus $(0,0)$, Directrix $x = 6$, Axis $y = 0$

(c) Vertex $(1,-2)$, Focus $(2,-2)$, Directrix $x = 0$, Axis $y = -2$

Q40

(a) $y^2 = 12x$

(b) $(x-2)^2 = 8(y+1)$

Q41

(a) Centre $(0,0)$, $a=5,b=3$ (horizontal), $c=4$. Vertices $(\pm 5,0)$, Foci $(\pm 4,0)$

(b) Centre $(1,-2)$, $a=4,b=5$ (vertical), $c=3$. Vertices $(1,3),(1,-7)$, Foci $(1,1),(1,-5)$

(c) $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Centre $(0,0)$, $a=3,b=2$, $c=\sqrt{5}$. Vertices $(\pm 3,0)$, Foci $(\pm \sqrt{5},0)$

Q42

(a) $\frac{(x-1)^2}{4} + \frac{(y+2)^2}{1} = 1$. Centre $(1,-2)$, Foci $(1 \pm \sqrt{3}, -2)$

(b) $\frac{(x-1)^2}{4} + \frac{(y+2)^2}{9} = 1$. Centre $(1,-2)$, Foci $(1, -2 \pm \sqrt{5})$

Q43

(a) $\frac{x^2}{25} + \frac{y^2}{9} = 1$

(b) $\frac{(x-2)^2}{9} + \frac{(y+1)^2}{25} = 1$

Q44

(a) Centre $(0,0)$, horizontal. $a=3,b=4,c=5$. Vertices $(\pm 3,0)$, Foci $(\pm 5,0)$, Asymptotes $y = \pm \frac{4}{3}x$

(b) Centre $(-2,1)$, vertical. $a=2,b=3,c=\sqrt{13}$. Vertices $(-2,3),(-2,-1)$, Foci $(-2,1\pm\sqrt{13})$, Asymptotes $y-1 = \pm \frac{2}{3}(x+2)$

Q45

$\frac{(x-1)^2}{4} - \frac{(y+2)^2}{9} = 1$. Centre $(1,-2)$, $a=2,b=3,c=\sqrt{13}$. Vertices $(3,-2),(-1,-2)$, Foci $(1\pm\sqrt{13},-2)$

Q46

(a) $\frac{x^2}{4} + \frac{y^2}{9} = 1$ — Ellipse

(b) $y = x^2 - 2x - 1$ — Parabola

(c) $xy = 1$ — Rectangular hyperbola

Q47

(a) $\frac{dy}{dx} = \frac{3t}{2}$, $\frac{d^2y}{dx^2} = \frac{3}{4t}$

(b) $\frac{dy}{dx} = -\cot t$, $\frac{d^2y}{dx^2} = -\csc^3 t$

Q48

$$3x - 2y = 1$$

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SECTION E: PART A SPEED ROUND — SOLUTIONS

Solutions E

Q49

(a) $I = e^{\int P(x),dx}$

(b) $\frac{n(n+1)}{2}$

(c) $c^2 = a^2 - b^2$ (ellipse)

(d) $c^2 = a^2 + b^2$ (hyperbola)

(e) $(h, k + a)$

Q50

(a) FALSE (b) TRUE (c) FALSE (d) FALSE (e) FALSE (f) FALSE